Abstract
We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg–de Vries type), geometry of Liouville-type hyperbolic systems (including the 2D Toda chains), and Euler–Lagrange gauge theories (such as the Yang–Mills theories, gravity, or the Poisson sigma-models). Also, we formulate several open problems.
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